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Theoretical (2020) 13:519–536 https://doi.org/10.1007/s12080-020-00465-8

ORIGINAL PAPER

Threshold harvesting as a conservation or exploitation strategy in management

Frank M. Hilker1 · Eduardo Liz2

Received: 8 May 2019 / Accepted: 18 May 2020 / Published online: 20 June 2020 © The Author(s) 2020

Abstract Threshold harvesting removes the surplus of a population above a set threshold and takes no harvest below the threshold. This harvesting strategy is known to prevent while obtaining higher yields than other harvesting strategies. However, the harvest taken can vary over time, including seasons of no harvest at all. While this is undesirable in or other exploitation activities, it can be an attractive feature of management strategies where removal interventions are costly and desirable only occasionally. In the presence of population fluctuations, the issue of variable harvests and population sizes becomes even more notorious. Here, we investigate the impact of threshold harvesting on the dynamics of both and harvests, especially in the presence of population cycles. We take into account semelparous and iteroparous life cycles, Allee effects, observation uncertainty, and demographic as well as environmental stochasticity, using generic mathematical models in discrete time. Our results show that threshold harvesting enhances multiple forms of population stability, namely persistence, constancy, resilience, and dynamic stability. We discuss plausible choices of threshold values, depending on whether the aim is exploitation, pest control, or the stabilization of fluctuations.

Keywords Fixed escapement · Harvest control rule · Pest control · Maximum sustainable yield · Intervention frequency · Population cycles · Stability

Introduction levels, detection thresholds, or values that can be used to justify spending a budget.1 The management of often requires population The threshold is sometimes also called the escapement harvesting. Examples include the control of invasive , (or reference point in the fisheries literature), removal of weeds, roguing of infected plants, regulation because it corresponds to the population level that “escapes” of pests such as defoliating insects, and the harvesting of harvesting. As a harvesting strategy that keeps the pop- fisheries and forestry. The strategy of threshold harvesting ulation at a constant ceiling, threshold harvesting can be removes the entire excess of a population stock above a seen as the antipode to the constant-quota harvesting strat- certain threshold value and takes no harvest if the stock egy, which removes a constant quota from the population. is below the threshold. In many instances, this strategy Proportional harvesting, which removes a constant fraction emerges naturally, e.g., when pests exceed economic injury of the population, corresponds to an intermediate strat- egy between the two extremes of threshold harvesting and constant-quota harvesting. Figure 1 illustrates the three har-  Frank M. Hilker vesting strategies by showing their relative harvest mortality [email protected] 1Equivalent effects on a population can be caused by mechanisms Eduardo Liz other than harvesting. The earliest examples we are aware of are given [email protected] by Ricker (1952) (and reiterated in his famous paper Ricker 1954, Fig. 34 and pp. 609–616). He describes the situation when a population is regulated by generalist predators in such a way that only individuals 1 Institute of Mathematics and Institute of Environmental in refuges of a given size escape or that predators switch their Systems Research, Osnabr¨uck University, 49069 Osnabr¨uck, search image abruptly at a certain prey threshold density. He refers to Germany two empirical examples, one being predation upon bob-white in Iowa 2 Departamento de Matematica´ Aplicada II, Universidade and the other one being predation upon Atlantic salmon parr in some de Vigo, 36310 Vigo, Spain rivers. 520 Theor Ecol (2020) 13:519–536

Threshold harvesting is known under many different names in quite different disciplines (see Table 1). Here, we use the term “threshold harvesting” because it seems the most inclusive one for the wide range of applications that we have in mind, stretching from pest control to resource exploitation. To avoid confusion, we point out that the term “threshold harvesting” is sometimes also used for other harvesting strategies, where there are different forms of removal above the threshold (e.g., Kaitala et al. 2003; Enberg 2005). According to a review of harvesting strategies, threshold harvesting generally performs best for maximizing cumu- lative yield, mean annual yield, and profit in the absence of observation error (Deroba and Bence 2008). This has been found to hold under a wide range of assumptions (e.g., Ricker 1958;Clark1976; Reed 1978, 1979, 1980; Ludwig 1979, 1980, 1998; Mendelssohn 1979; Mangel 1985; Lande et al. 1995, 1997). As harvesting is curtailed when the pop- ulation is below the threshold level, threshold harvesting Fig. 1 Relative harvest mortalities for threshold harvesting (blue, has a built-in mechanism to accommodate random varia- dotted), proportional harvesting (black, dashed), and constant-quota harvesting (red, solid). We denote by x the population size at tion in environmental conditions or vital rates (Lande et al. equilibrium, Y(x) the corresponding yield, T the threshold, c the 1997). This has two major consequences. On the one hand, constant quota, and q the proportional rate of harvest this makes threshold harvesting a precautionary strategy that prevents overexploitation and maintains spawning biomass. On the other hand, threshold harvesting may impose a “stop- (or per-capita harvest rate) as a function of population and-go ” (Evans 1981), which implies an economic . For threshold harvesting, the relative population and political cost to more precautionary harvesting. Ricker removal ceases as the population becomes smaller and stops (1958) was perhaps the first to observe that the variability altogether when the population drops below the thresh- of catch is greater for threshold harvesting than for some old. By contrast, for constant-quota harvesting, the relative other strategies (see also Larkin and Ricker 1964;Tautz population removal increases as the population becomes et al. 1969; Allen 1973; Gatto and Rinaldi 1976). Another smaller. For proportional harvesting, the relative population drawback of threshold harvesting is that it requires up-to- removal is constant. date knowledge of stock size (Hilborn and Walters 1992).

Table 1 A collection of the many names of threshold Name References harvesting Ecology, fisheries, natural resources management Threshold harvesting Lande et al. (1995) and Lande et al. (1997) Fixed escapement Hilborn (1979) and Fryxell et al. (2005) Constant escapement Reed (1979) and Getz and Haight (1989) Fixed stock Jonzen´ et al. (2002) Constant-stock-size strategy Hilborn and Walters (1992) Optimal strategy Ludwig (1998) Bang–bang Clark (1976) Optimum stabilization of escapement Ricker (1958) Trigger harvest Sabo (2005) (upper) limiter control Hilker and Westerhoff (2006) and Tung et al. (2014) Physics, telecommunication, nonlinear dynamics Thresholding Sinha (1994) and Sinha (2001) Flat-topped maps Glass and Zeng (1994) Simple limiter control Corron et al. (2000) and Stoop and Wagner (2003) Theor Ecol (2020) 13:519–536 521

Errors in stock size estimates may (Engen et al. 1997; account stochasticity in the form of demographic and auto- Milner-Gulland et al. 2001; Lillegard˚ et al. 2005)ormay correlated environmental variations as well as observation not (Butterworth and Bergh 1993; Polacheck et al. 1999) error, and investigates the harvest frequency and the vari- change the relative superiority of threshold harvesting com- ability of yield as a function of harvesting intensity. When pared to other strategies. In practice, thresholds are typically we say increased threshold harvesting we mean setting a set too low (Ludwig et al. 1993). Yet, it remains that thresh- lower threshold (escapement) which amounts to elevated old harvesting is a strategy minimizing risk while harvesting intensity. In each of these sections, we not only optimizing yield (Lande et al. 1995; Lande et al. 1997) present results, but also place the models in the context and could thus bring together conservation and exploitation of the literature and discuss the relevance of our findings. interests. “Conclusions” provides a summary of the results and draws Recent microcosm experiments have investigated thresh- overall conclusions. old harvesting of of the ciliate Tetrahy- mena thermophila (Fryxell et al. 2005) and the fruit fly Drosophila melanogaster (Tung et al. 2016), for selected Population dynamic effects of threshold threshold values. Both studies support the utility of thresh- harvesting old harvesting as a management strategy. In both experi- ments, threshold harvesting was found to reduce variability In this Section, we study single-species populations without in and to reduce extinction risk. managed by threshold harvesting. Let xn be Depending on the context, harvesting may be desirable the population size at time step n, n = 0, 1, 2,.... because it generates revenue (e.g., in fisheries) or unde- The in the absence of harvesting is sirable because it causes expenditures (e.g., in pest con- assumed to follow trol). In this paper, we focus on three areas of application, x + = f(x ) (2.1) namely exploitation, pest control, and the dynamic stabi- n 1 n lization of fluctuating populations. Populations with fluctu- for an initial value x0 > 0. The map f gives the ations present unique challenges to managers (Barraquand discrete-time reproduction. In this Section, we assume a et al. 2017); for instance, the harvest can vary dramatically typical condition for f , which is satisfied by semel- and between years and sometimes stop completely. While the iteroparous population models without Allee effect, such as latter is desirable for pest control, it can be hazardous for the Beverton–Holt, Ricker, Hassell or Ricker–Clark maps: fishing industries. We will consider population fluctuations that are endogenously generated (due to overcompensation, (A) f :[0, ∞) →[0, ∞) is continuous, has a unique time lags, or nonlinear population interactions), including positive fixed point K, f(x)>x for x ∈ (0,K),and the case where there are additionally exogenous fluctua- 0 K. tions (due to random variations in the and in the If the population is managed by threshold harvesting, physical environment). This differs from previous work then the dynamics is governed by the difference equation by Lande and colleagues (reviewed in Lande et al. 1997),  which considers fluctuations in dynamically sta- ≤ = { }:= = f(xn) if f(xn) T, ble populations and assumes the absence of endogenously xn+1 min f(xn), T g(xn) T if f(xn)>T, generated oscillations. (2.2) Harvesting is perceived as magnifying exogenous and reduc- ing endogenous fluctuations (Milner-Gulland and Mace where T>0 is the threshold. The smaller the value of T , 1998). For a more informed basis, we consider different the larger the harvesting intervention. The graph of Eq. (2.2) forms of stability, namely persistence stability (persisting is a truncated map as illustrated in Fig. 2a, c. through time as opposed to going extinct); constancy sta- Next, we will state a few properties regarding the dynamic bility (population size remaining essentially unchanged); stability of a managed population, depending on whether the dynamic stability (eventually returning to the equilibrium population is stable or unstable in the absence of harvesting. after perturbations); and resilience (the ability to absorb perturbations that are within the domain of attraction) Dynamic stability (Grimm and Wissel 1997). In the next Section, we inves- tigate the effect of threshold harvesting on these forms of Here, we will show that threshold harvesting can never stability for semelparous and iteroparous species. “Models have a destabilizing effect on the managed population. with Allee effect” considers populations with demographic By destabilizing, we mean that an equilibrium that is Allee effect. “Average long-term yield” focuses on the locally asymptotically stable in the absence of harvesting average long-term yield. “Stochastic simulations” takes into becomes unstable due to harvesting. Instead, we will show 522 Theor Ecol (2020) 13:519–536

Fig. 2 Examples of threshold 120 120 harvesting with T = 50 (top a b row) and T = 80 (bottom row). In both cases, the unmanaged 100 100 population cycles with period 4, 1 + following a Ricker map with n 80 80 r = 2.6 and K = 60. Diagrams in the left column show the unmanaged Ricker map (hump- 60 60

shaped) and the harvested Harvest population maps (flat-topped). 40 40 Threshold harvesting begins Population size after ten time steps. For the low Population size, x 20 threshold (top row), harvesting 20 stabilizes the population on a 0 stable equilibrium (b), shown as 0 red point in (a). For the large 0 20 40 60 80 100 120 0 5 10 15 20 25 threshold (bottom row), Population size, xn Time, n harvesting changes the period of 120 120 the from 4 to 2 c d (d), shown in thin dotted line in (c) 100 100 1

+ n 80 80

60 60 Harvest

40 40 Population size Population size, x

20 20

0 0 0 20 40 60 80 100 120 0 5 10 15 20 25

Population size, xn Time, n that strong enough harvesting, i.e., a small escapement, First, consider the dynamically simplest case. So we stabilizes population dynamics. assume that the positive equilibrium K is a global attractor In fact, if the threshold is set below the of Eq. 2.1. The following result states that this equilibrium (T 0 global attractor of Eq. (2.1).IfT ≥ K,thenK is the unique converge to K. If this property holds, we say that K is positive equilibrium of Eq. (2.2), and it is a global attractor. a global attractor. We state this property in the following proposition, whose proof can be found in Appendix A.1: This result is a particular case of a more general result proved in Hilker and Liz (2019, Proposition A.3). Proposition 2.1 Assume that f satisfies (A).IfT ≤ K, Second, consider that the equilibrium K of Eq. (2.1)is then T is the unique positive equilibrium of Eq. (2.2), and unstable and that Eq. (2.1) has a finite number of periodic it is a global attractor. orbits. We will show that increased harvesting pressure, i.e., decreasing the threshold, induces a sequence of In the remainder of this section, we consider the scenario period-halving bifurcations until the equilibrium becomes that the threshold is set at a value greater than the carrying stable. It is possible to determine the threshold value for capacity (T>K). Then, the dynamics of the managed desired periods, which includes the threshold necessary system (2.2) depends on the dynamics of the unmanaged for stabilization. Figure 2c and d show an example where system (2.1). We distinguish three cases, namely where the threshold harvesting changes a 4-cycle into a 2-cycle. unmanaged population (i) has a stable equilibrium, (ii) is An application of Sharkovsky’s theorem on the coexis- periodic, or (iii) chaotic. tence of cycles (see Sharkovsky et al. 1997) ensures that Theor Ecol (2020) 13:519–536 523 all the periods are powers of two. For the unimodal repro- duction maps usually employed in population dynamics, if p = 2N is the largest period of a periodic orbit of f ,then this p-periodic orbit is unique, and it attracts all initial con- ditions except those belonging to the other periodic orbits and their preimage. Denote the attracting periodic orbit by O ={x1,x2,...,xp},wherex1 < ··· Kis equivalent to f(T) > T.If threshold value T , ranging between 0 and 140 (a little bit beyond the ≤ 2 maximum of f ). The uncontrolled map f has an attracting cycle of f(T)

(or perennial) species that can have multiple reproductive results, threshold harvesting does not destabilize the positive cycles over their lifetime. Iteroparous species are described equilibrium (Fig. 4c). The corresponding models are by bimodal maps, for which both constant-quota and pro- = − portional harvesting can be destabilizing (see below). xn+1 (1 γ/M)f(xn) (proportional harvesting), Although proportional harvesting tends to simplify the xn+1 = max{f(xn) − d,0} (constant-quota harvesting), dynamics for a unimodal Ricker-type model, it has been xn+1 = min{f(xn), T } (threshold harvesting), recently shown that this is not always the case for the Ricker–Clark model (Clark 1976;Thieme2003; Liz 2010) where M ≈ 141 is the maximum of f .

− Constancy and persistence stability f(x)= αx + (1 − α)x er(1 x/K), (2.4) One measure of constancy stability is the fluctuation range, where the survival parameter α describes survivorship of and one measure of persistence stability is the inverse of adults from one season to the next. Equation 2.4 has been the minimum population size (as extinction risk is larger the recently employed by Shelton and Mangel (2011) to study smaller the population size). On the basis of these measures, how fishing can magnify fluctuations in fish populations, we find that increased threshold harvesting can enhance, andbyYakubuetal.(2011) to asses the performance of but that it never deteriorates constancy stability. Moreover, proportional harvesting. Even if the survival rate is very we find that increased threshold harvesting deteriorates small, a 2-periodic attractor of Eq. (2.4) can be transformed persistence stability for TK. and Ruiz-Herrera 2012,Fig.8).Forlargervaluesofα, As we have mentioned, typical unimodal population a globally attracting equilibrium can be destabilized by maps (e.g., Ricker, Hassell, Maynard–Smith, quadratic) harvesting. For proportional harvesting, this phenomenon have a unique positive equilibrium K and are decreasing has been noticed by Yakubu et al. (2011) and analyzed in for x>K. In this case, if the equilibrium is unstable, a detail by Liz and Ruiz-Herrera (2012); for constant-quota common feature of the bifurcation diagrams in the presence harvesting, this effect has been shown in Liz (2010). In view of threshold harvesting (2.2) is that, once the harvesting of Propositions 2.1 and 2.2, threshold harvesting cannot is implemented, the enveloping curves of the diagram are destabilize a global attractor. For comparison, we consider defined by the line x = T and the curve x = f(T)until the Eq. 2.4 with K = 60, r = 4, and α = 0.56. In this equation has a stable fixed point at T = K (for example, case, the equilibrium K is a global attractor. A strategy see Fig. 3). The reason is that, since the periodic attractor  of proportional harvesting destabilizes the equilibrium by has the form (2.3), it is clear that the maximum value of  a period-doubling bifurcation (Fig. 4a). Constant-quota is T and the minimum is f(T). This has two immediate harvesting not only destabilizes the equilibrium, but also implications. First, the extent of the periodic attractor (i.e., induces a strong Allee effect capable of producing essential the range of fluctuations) is an increasing function of T extinction or a sudden collapse if the catch quota is large (T>Kimplies that f (T ) < 0 and hence T − f(T) is enough (Fig. 4b). Finally, in accordance with our analytical increasing); in other words, constancy stability increases as

abc

Fig. 4 Bifurcation diagrams for the bimodal Ricker–Clark map (2.4) strong Allee effect, and that there are boundary crises (Schreiber managed by three different harvesting strategies. a For proportional 2001). c Threshold harvesting does not destabilize the equilibrium. harvesting, a bubble is created (cf. Liz and Ruiz-Herrera 2012). b For Red dashed lines indicate unstable equilibria. Initial conditions drawn constant-quota harvesting, the equilibrium is destabilized too; note from a pseudo-random uniform distribution. Parameter values: K = that an additional unstable equilibrium appears due to the induced 60, r = 4, α = 0.56. M ≈ 141 is the maximum value of f Theor Ecol (2020) 13:519–536 525 the intensity of threshold harvesting increases. This property a combination of overcompensatory instability and the is no longer true for bimodal maps if the cycle has a strong Allee effect. Thus, it is natural that any harvesting period larger than two. Second, the minimum population method can prevent essential extinction when it reduces size of the periodic attractor, i.e. f(T), increases with lower the undesirable effects of overcompensation. The transition values of T as long as T>K. As stochastic extinction is from essential extinction to the existence of a nontrivial more likely for small population sizes, increased threshold attractor takes place via a boundary collision. Some harvesting enhances persistence stability for T>K.If examples for constant-quota harvesting and proportional T ≤ K and the population is dynamically stabilized to harvesting can be found in Sinha and Parthasarathy (1996), equilibrium, this is no longer the case. Schreiber (2001), Liz (2010), and Cid et al. (2014). Threshold harvesting prevents essential extinction, too. Moreover, the escapement level necessary to avoid essential Models with Allee effect extinction is very easy to compute: just take the value of T>K2 such that f(T) = K1, that is, the preimage In this section, we investigate the effect of threshold harvest- of K1 by the map f . If the escapement is set too low, ing on population dynamics with a strong Allee effect (also then the species is doomed to extinction through a sudden called critical in the fisheries literature) in the collapse induced by the Allee effect; however, compared absence of harvesting. So far, Allee effects are acknowl- with constant-quota or proportional harvesting, this effect edged to increase extinction risk and to reduce yield as well is less dramatic for threshold harvesting since this is only as yield predictability under threshold harvesting (Ricker possible if the escapement is below the Allee threshold K1. 1958; Collie and Spencer 1993; Eggers 1993; Lande et al. We provide some examples to illustrate these remarks. 1995; Walters and Parma 1996; Ludwig 1998). Here, we First, consider Eq. (3.1) with K = 60 and r = 5. will show that threshold harvesting is a remarkably resilient Figure 5a shows the stability diagram for T ∈ (0, 200) and and adaptive strategy in comparison to constant-quota and s ∈ (0.01, 0.04). The border between essential extinction proportional harvesting and that it can prevent essential and bistability is defined by the implicit curve f(T) = K1 extinction, in which almost all initial conditions would lead (T>K2); the equilibrium K2 is attracting when K1 < to extinction in the absence of harvesting. T ≤ K2; and there is extinction for T ≤ K1. These generic We consider the Ricker-Schreiber model dynamics can be observed in Fig. 5b for the particular value sx n r(1−xn/K) = xn+1 = xn e := f(xn), n = 0, 1, 2,..., (3.1) s 0.025 (corresponding to the dashed line in Fig. 5a). 1 + sxn Essential extinction can be prevented if T0 represents an individual’s efficiency and there is a nontrivial attractor if T2

Fig. 5 a Stability diagram of the Allee effect model (3.1) ab managed by threshold harvesting in the parameter plane (T , s) with K = 60 and r = 5. b Bifurcation diagram for fixed s = 0.025 (corresponding to the horizontal dashed line in a). Initial conditions drawn from a pseudo-random uniform distribution 526 Theor Ecol (2020) 13:519–536

Fig. 6 Bifurcation diagrams (top row) and asymptotic yield (bottom row) for different harvesting strategies and the Ricker–Schreiber model (3.1) with K = 1,r = 4,s = 0.6. For threshold harvesting, intervention starts at T ≈ 0.844, the maximum value of f .Inthe top row, initial conditions are drawn from a pseudo-random uniform distribution. The bottom row is for an initial condition not leading to extinction

and Allee effects. Before a collapse occurs, there is an basin of attraction of the extinction state markedly expands attracting equilibrium at which the population stabilizes due to these harvesting strategies. By contrast, the Allee its size for intermediate initial population values; higher threshold remains constant for increased threshold harvest- values of this equilibrium make the collapse more dramatic, ing. As a consequence, under threshold harvesting, the since the manager does not have early signals indicating the population is more resilient against perturbations than that impending collapse. In our example, we can compute the under the other harvesting strategies. population size at the bifurcation point leading to extinction. The asymptotic values of the average sustained yield is shown in Fig. 6. For constant-quota and proportional harvest- 1. For constant-quota harvesting, a collapse occurs if we ing, the average yield collapses at or respectively shortly harvest at a constant quota of d = 0.4678. The after the harvesting intensity producing maximum sustain- equilibrium at the bifurcation point is K (d) ≈ 0.35, 2 able yield. For threshold harvesting, there is no abrupt which corresponds to 50% of the carrying capacity in collapse of the yield; it changes gradually even beyond the absence of harvesting (K ≈ 0.69). 2 the maximum sustainable yield and despite a strong Allee 2. For proportional harvesting, a collapse occurs if we effect. This is a desirable feature for the adaptive man- harvestatarateofγ = 62% of the population, and the agement of developing fisheries (see Hilborn and Walters equilibrium at the bifurcation point is K (γ ) ≈ 0.22, 2 1992). In the next section, we will investigate the shape of which corresponds to 31.8% of the carrying capacity in the yield curve for threshold harvesting in more detail. the absence of harvesting. 3. Finally, for threshold harvesting, a collapse occurs for = ≈ T K1 0.036, which is also the population Average long-term yield size of the nontrivial attractor at the bifurcation point and corresponds to 5% of the carrying capacity in the An important feature of threshold harvesting (2.2)isthat absence of harvesting. in many cases there is a periodic attractor which attracts The Allee threshold corresponds to a tipping point of pop- all positive orbits with probability one. Moreover, the orbits ulation sizes above which populations survive and below become periodic after few transients. Thus, after these which populations go extinct. For constant-quota and pro- transients have died out, intervention takes place only once portional harvesting, the Allee threshold sharply increases every p years, where p is the period of the attracting cycle. with increasing harvesting intensity, especially near the This property provides a simple method to calculate the bifurcation points (red dashed curves in Fig. 6). That is, the frequency of harvest and the average long-term yield. Theor Ecol (2020) 13:519–536 527

In this Section, we consider again threshold harvest- (e.g., Hall et al. 1988; Lande et al. 1995). Here, we find the ing (2.2) of a population with the generic condition (A), i.e., existence of additional dome-shaped relationships between without Allee effect. Assume that average long-term yield and threshold in parameter ranges p−1 T>Kwhen the population dynamics is unstable in the p ={T,f(T),...,f (T )} absence of harvesting. To our knowledge, these additional i is the attracting cycle. Notice that f (T ) < T holds for local maxima in sustained yield have not been reported all i = 1, 2,...,p− 1. Then, harvesting occurs only once before for threshold harvesting (but see Hilker and Liz 2019 every p time steps, and what we remove from the population for the strategy of proportional threshold harvesting). (harvest, yield) is Figure 7 also shows the mean population size. We briefly f p(T ) − T . report that there is a hydra effect (Abrams 2009; Hilker and Liz 2013)forT ∈ (60, 81.4), approximately. That is, the ≤ = For example, if T K,thenp 1 such that there mean population size increases when harvesting mortality is is an intervention every time step, and the average yield is enhanced within this parameter range. f(T)− T . If f is differentiable, then it is easy to compute the threshold values that give the maximum long-term yield. Stochastic simulations Indeed, for a p-periodic window, we have to calculate the = p − maximum of gp(x) f (x) x, which is reached at There is intrinsic uncertainty in population dynamics, assess- = the point T such that gp(T ) 0, which is equivalent to ment, and harvesting. This uncertainty is known to be prob- = = ··· μ 1, where μ f (q1)f (q2) f (qp) is the multiplier lematic for harvesting and sustainable resource use (Hine { } associated to the cycle q1,q2,...,qp . For example, let us and Gifford 1996; Roughgarden and Smith 1996; Gustafs- = = consider the Ricker map with r 2.6 and K 60 (Fig. 7). son et al. 1999; Adamson and Hilker 2020). In this section, ∈ In the range T (0, 60), T is an attractor and the maximum we consider three sources of random variation, namely demo- ≈ ≈ harvest is attained at T1 19.2, with a yield value h1 graphic and environmental stochasticity (chance events and ∈ 93.3. The 2-periodic window is T (60, 105.1),andthe variation of external conditions) as well as observation error maximum yield within this threshold range is attained at (generated by the measurement process). T2 ≈ 84, with a value of h2 ≈ 26.4. However, since in this case harvesting occurs only once every two periods, the Model description maximum average yield is h2/2 ≈ 13.2. ≤ That is, for threshold values T K, the average long- Environmental stochasticity enters each generation in the term yield shows a dome-shaped form, which has been form of a first-order autoregressive model: reported before for different types of population models  + = ρεn σenv δn+1 for n>0 εn+1 = (5.1) 100 100 δn+1 for n 0 Mean pop. size Mean yield where  80 80 − 2 − ∼ σenv × 1 ρ 2 δn Normal ,σenv (5.2) 2 1 − ρ2 60 60 is an uncorrelated, normally distributed process error with a mean that corrects the bias from the autocorrelation in the 40 40 Harvest environmental variation (Thorson et al. 2014). Parameter

Population size σenv > 0 determines the magnitude of environmental ∈ 20 20 variation. ρ (0, 1) is the strength of the autocorrelation. If ρ = 0, there is no autocorrelation and the noise is white. If ρ is strictly positive (negative), the noise spectrum is reddened 0 0 140 120 100 80 60 40 20 0 (blue-shifted). Here, we assume temporally autocorrelated Threshold, T environmental variations, as this has been considered important in ecosystems (Steele 1985) and for the choice Fig. 7 Bifurcation diagram with the long-term harvest (thin red), of management strategies (Hall et al. 1988;Koslow1989; average long-term yield (bold red), and mean population size (solid black). Threshold harvesting strategy (2.2) with the same Ricker map Walters and Parma 1996; Kaitala et al. 2003). The individual as in Fig. 3 reproduction is drawn from a Poisson distribution with a 528 Theor Ecol (2020) 13:519–536 mean given by the reproduction curve f(xn), modulated We do not explicitly incorporate implementation error (also by the log-normally distributed environmental noise. The called partial controllability) but note that the stochastic population dynamics before harvesting is then described by harvest may be considered to subsume both observation

εn and implementation error (cf. Jonzen´ et al. 2002). For pro- x˜n = Poisson f(xn)e , (5.3) portional harvesting, Hn = γ x˜obs,n. For constant-quota which gives integer values as population abundance. harvesting, Hn = d is assumed to be independent of The population size in the next generation population size estimates. x + =˜x − H (5.4) n 1 n n Environmental stochasticity is the one after harvesting, where the harvest in the current generation is given by In the following Monte–Carlo simulations, we keep the  x˜ − T if x˜ >T autocorrelation in the environmental variation at a fixed H = obs,n obs,n (5.5) = n 0else value of ρ 0.43, which is the mean from a meta- analysis of 154 (Thorson et al. 2014). For the and based on the observed population size  moment, we ignore observation error but will investigate 2 its impact separately later on. We have varied the strength σobs x˜obs,n =˜xn exp σobs ωn − , (5.6) of environmental stochasticity. Figure 8 shows results for 2 three cases. The intermediate one is σenv = 0.74, which where ωn is drawn from a standard normal distribution. That corresponds to the mean from the fish stock meta-analysis is, the harvest is stochastic as it includes observation uncer- (Thorson et al. 2014). As fish populations tend to show tainty (also called partial observability), which is assumed substantial environmental variation in their , we to be log-normally distributed (Hilborn and Mangel 1997). also consider σenv = 0.4 as a lower value (which may still

Fig. 8 Comparison of threshold 4 1 4 1 K harvesting (left column) and K a 0.8 b 0.8 proportional harvesting (right 3 3 column) for different levels of 0.6 0.6 environmental stochasticity (first 2 2 row σenv = 0.4, second row 0.4 0.4 = = σenv 0.74, third row σenv 1). 1 1 Shown are the coefficient of 0.2 0.2 variation of the harvest, the CV harvest, Harvest/ 0 0 CV harvest, Harvest/ 0 0 mean harvest scaled by the 140 120 100 80 60 40 20 0 0 0.2 0.4 0.6 0.8 1 Extinction risk, Harvest freq. carrying capacity, the extinction Extinction risk, Harvest freq. Threshold, T Harvest proportion, γ risk, and the harvest frequency. 4 1 4 1 K Model Eqs. (5.1)–(5.6) with the K c Ricker map for r = 2.6, 0.8 d 0.8 K = 60, ρ = 0.43 including 3 3 demographic stochasticity but 0.6 0.6 2 2 no observation error (σobs = 0). Results are based on 10,000 0.4 0.4 1 1 replicates and 100 iterations for 0.2 0.2 each harvesting intensity, with CV harvest, Harvest/ the initial 50 iterations discarded CV harvest, Harvest/ 0 0 0 0 to remove transient effects. 140 120 100 80 60 40 20 0 0 0.2 0.4 0.6 0.8 1 Extinction risk, Harvest freq. Extinction risk, Harvest freq. Random initial conditions Threshold, T Harvest proportion, γ 4 1 4 1 K K e 0.8 0.8 3 3 f 0.6 0.6 2 2 0.4 0.4 1 1 0.2 0.2

CV harvest, Harvest/ 0 0 CV harvest, Harvest/ 0 0

140 120 100 80 60 40 20 0 0 0.2 0.4 0.6 0.8 1 Extinction risk, Harvest freq. Extinction risk, Harvest freq. Threshold, T Harvest proportion, γ

CV harvest Harvest/K Extinction risk Harvest freq. Theor Ecol (2020) 13:519–536 529 be considered high for other taxa or certain species) and there would be no harvesting in deterministic systems. σenv = 0.9 as a higher value. For larger environmental stochasticity, the noise smears We would like to highlight the following observations in out the deterministic pattern even more, so that even low Fig. 8. First, consider extinction risk, which we define as the escapements do not guarantee harvest in each generation proportion of simulation replicates leading to extinction (Fig. 8f). (xt < 1) over the time horizon of 100 generations. Increased Proportional harvesting occurs in each generation, threshold harvesting, i.e., a lower escapement, decreases the provided the population has not gone extinct. The earlier extinction risk (solid red curves in Fig. 8a, c, e). This is a population goes extinct, the lower the harvest frequency. because threshold harvesting curtails large population sizes This is why proportional harvesting does not always have a that could trigger an overcompensatory population response 100% harvest frequency in the stochastic models (Fig. 8b, and generate vulnerably small population sizes in the next d, f). Note that the frequency of proportional harvesting is generation. We note that the inverse of extinction risk is overall higher than that of threshold harvesting for small another measure of persistence stability, which suggests that environmental stochasticity (compare Fig. 8a and b), but smaller escapements enhance persistence stability. For exces- that it is generally lower than that of threshold harvesting sively small escapement levels close to nil, however, the for large environmental stochasticity (compare Fig. 8e extinction risk rises sharply due to plain overexploitation. and f). This runs counter to the usual notion that threshold Proportional harvesting implies an elevated extinction harvesting implies more harvest closures. The reason is that risk compared to threshold harvesting (see Fig. 8b, d, f). threshold harvesting can sustain populations that would go This is because proportional harvesting is applied at all pop- extinct under proportional harvesting. ulation sizes, even if they are already small and will be fur- Third, increased threshold harvesting can reduce the ther reduced by proportional harvesting. For large environ- variability in yield, measured by its coefficient of variation mental stochasticity, almost all simulated populations have (solid black curves in Fig. 8). In other words, the gone extinct under proportional harvesting (Fig. 8f). For predictability of the catch increases when setting lower small and medium environmental stochasticity, increased escapements; this holds for a wide range of escapement proportional harvesting can reduce extinction risk provided values with an exception near the carrying capacity for low it is not too intense. This is because proportional harvesting environmental stochasticity (Fig. 8a), as the deterministic stabilizes the population oscillations. However, if the har- population would never be harvested at this equilibrium. For vesting proportion is too large, the extinction risk rises again larger environmental stochasticity, the harvest variability due to overexploitation (Fig. 8b, d). This may occur well decreases throughout with lower escapements (Fig. 8c, before maximum sustainable yield is achieved (Fig. 8d) and e). However, the relative variation in yield increases in over a wider parameter range than for threshold harvesting. the unlikely case that the escapement approaches nil so Hence, proportional harvesting seems inferior in promot- that the population would be harvested deterministically to ing persistence stability compared to threshold harvesting. extinction. For constant-quota harvesting, basically all simulated pop- For proportional harvesting, yield variability increases ulations have gone extinct, which is why the results for this with harvesting intensity when the latter is high. For low harvesting strategy are not included in this Section. and medium harvesting intensities, increased proportional Second, we observe that harvest frequency tends to harvesting reduces yield variability. This is in contrast increase with increased threshold harvesting, i.e., for with the classical result that yield variability increases with lower escapements (see dashed red curves in Fig. 8a, c, increased harvesting intensity for logistically growing pop- e). For small environmental stochasticity, harvest frequency ulations in random, uncorrelated environments (Beddington follows the deterministic staircase pattern predicted by our and May 1977). The reason for this difference may be that analytical results in “Average long-term yield,” but with the increased proportional harvesting stabilizes the endogenous sharp transitions smoothened out by the random variations population oscillations generated by the overcompensatory (Fig. 8a). The latter can have twofold effects. On the one population dynamics: the less variable the population size, hand, they lead to population sizes below the threshold the less variable the yield. so that there is no harvesting. This is why there may be Note that for both threshold and proportional harvesting occasionally no harvesting for T

Fig. 9 Comparison of threshold 4 1 4 1 K harvesting (left column) and K a 0.8 b 0.8 proportional harvesting (right 3 3 column) for different levels of 0.6 0.6 observation uncertainty (top row 2 2 σobs = 0.3, bottom row 0.4 0.4 = σobs 0.55). Shown are the 1 1 coefficient of variation of the 0.2 0.2 CV harvest, Harvest/ harvest, the mean harvest scaled CV harvest, Harvest/ 0 0 0 0 by the carrying capacity, the 140 120 100 80 60 40 20 0 0 0.2 0.4 0.6 0.8 1 Extinction risk, Harvest freq. extinction risk, and the harvest Extinction risk, Harvest freq. Threshold, T Harvest proportion, γ frequency. Model 4 1 4 1 K Eqs. (5.1)–(5.6) with the Ricker K c map for r = 2.6, K = 60, d 0.8 0.8 ρ = 0.43 including 3 3 demographic and environmental 0.6 0.6 stochasticity (σenv = 0.74). 2 2 Results are based on 10,000 0.4 0.4 1 1 replicates and 100 iterations for 0.2 0.2 each harvesting intensity, with CV harvest, Harvest/ the initial 50 iterations discarded CV harvest, Harvest/ 0 0 0 0 to remove transient effects. 140 120 100 80 60 40 20 0 0 0.2 0.4 0.6 0.8 1 Extinction risk, Harvest freq. Extinction risk, Harvest freq. Random initial conditions Threshold, T Harvest proportion, γ

CV harvest Harvest/K Extinction risk Harvest freq. Theor Ecol (2020) 13:519–536 531

Further increasing the observation error to a 59.4% Bjørnstad and Grenfell 2001; Fromentin et al. 2001; Shelton coefficient of variation (σobs = 0.55) affects the perfor- and Mangel 2011), we have considered a range of ecological mance of threshold harvesting marginally (compare Fig. 9c models and environmental conditions including iteroparity, with a). By contrast, the increased observation uncertainty strong Allee effects, and demographic and environmental causes almost always extinction for proportional harvest- stochasticity as well as observation uncertainty. Table 2 ing (Fig. 9d). This suggests that threshold harvesting may summarizes the main results of this paper. The emerging be more robust against observation uncertainty than propor- theme is that threshold harvesting tends to enhance multiple tional harvesting. stability dimensions (dynamic, constancy, and persistence stability) and seems more resilient and robust against abrupt collapses than proportional and constant-quota harvesting. Conclusions Our analytical results in this respect are quite general and apply to deterministic discrete-time single-species As it is now widely recognized that harvesting and intrinsic models. Our simulation results focus on overcompensatory population dynamics as well as environmental factors population dynamics that are oscillatory. Overcompensation jointly impact the size and variability of populations (e.g., has been observed in insects (Nicholson 1957; Costantino

Table 2 Summary of main results

Populations without Allee effect (analytical results for semelparous species satisfying condition (A))

• TH is never destabilizing (“Dynamic stability”)—this also holds for iteroparous species (“Bimodal maps with iteroparity”). • TH guarantees a globally asymptotically stable equilibrium for T ≤ K (Prop. 2.1) and the existence of period halvings when the carrying capacity is unstable for T>K; the latter result additionally requires f to be an SU map (“Dynamic stability”). • Constancy stability (measured by fluctuation range) always increases when T is lowered; this does not apply to iteroparous species with cycles of a period > 2(“Constancy and persistence stability”). • Persistence stability (measured by inverse of lowest population abundance) increases for T>Kand decreases for T

Populations with Allee effect (simulation results for the Ricker–Schreiber map)

• TH is able to prevent essential extinction (Fig. 5); the required threshold value is easy to find analytically (“Models with Allee effect”). • If a sudden collapse occurs under TH, this is less dramatic than for PH and CH (Fig. 6). • Allee threshold is only marginally increased by TH in comparison to PH and CH (Fig. 6). • Dome-shaped yield curve without abrupt collapse amenable for adaptive management (Fig. 6).

Yield (semelparous species satisfying condition (A))

• Analytical formulas of average long-term yield, including MSY, and long-term harvest frequency as a function of threshold values (for periodic systems; “Average long-term yield”) • There are additional local maxima in sustained average yield for T>K(“Average long-term yield”).

Random fluctuations (simulation results for model (5.1)–(5.6) with the Ricker map, r = 2.6,K = 60,ρ = 0.43)

• Lower escapements tend to enhance persistence stability (by reducing extinction risk), reduce harvest closures (i.e., increase harvest frequency), and increase yield predictability (by reducing harvest variability) (Figs 8 and 9). • The escapement level producing maximum sustainable yield comes with low extinction risk, almost no harvest closures, and is robust against levels of environmental variation and observation uncertainty (Figs. 8 and 9). • TH sustains the population in many cases where PH and CH lead to population collapse (Figs. 8 and 9).

TH threshold harvesting, PH proportional harvesting, CH constant-quota harvesting 532 Theor Ecol (2020) 13:519–536 et al. 1995), fish (Zipkin et al. 2008; Foss-Grant et al. 2016), dome-shaped relationship between yield and harvesting plants (Thrall et al. 1989; Buckley et al. 2001; Pardini et al. effort (here: lower thresholds) that is ubiquitously found in 2009), and ungulates (Grenfell et al. 1992). deterministic and stochastic models. This may be due to the When overcompensation produces small population limited number of harvesting efforts that were investigated. sizes, there is an increased risk of extinction, especially Tung et al. (2016) used two threshold values, which are in stochastic environments. In our stochastic simulations definitely too few to find a dome-shaped relationship. (limited to certain life-history and autocorrelation param- Fryxell et al. (2005) found a monotonic rather than a dome- eters), threshold harvesting emerges as a more sustainable shaped relationship, but the four threshold values that they strategy than proportional and constant-quota harvesting. used (corresponding to 100%, 75%, 50%, and 25% of the More specifically, threshold harvesting tends to sustain the carrying capacity) may have also been chosen at levels not population in conditions where proportional and constant- revealing the full relationship. quota harvesting lead to population extinction. The under- In the literature, threshold harvesting is known to not lying reason is that threshold harvesting is stopped below only produce higher yields than other harvesting strategies, the escapement and does not further reduce small popula- but also maintain larger and less variable population sizes tion sizes. Proportional and constant-quota harvesting lack in many circumstances (Ricker 1958; Gatto and Rinaldi a built-in protection of small population sizes that may 1976; Getz and Haight 1989; Lande et al. 1997; Deroba occur as a result of overcompensation, random variation, or and Bence 2008). However, this comes at the cost of observation uncertainty. increased yield variability, which is attributed to fishery If threshold harvesting is used as a strategy for reducing closures or ceased harvesting when the population size intrinsic population fluctuations, then a threshold value falls below the threshold (Lande et al. 1997; Lillegard˚ near the carrying capacity seems a good choice as this et al. 2005). The simulations in this paper suggest is predicted to maintain a mean population size similar that harvesting closures are unlikely around thresholds to the unharvested population and to have little burden producing MSY, as harvest frequency approaches 100% for in terms of having to harvest. This has been observed in thresholds of about 80% the carrying capacity or lower. deterministic (see Fig. 7; harvest frequency not explicitly The threshold level producing MSY and the MSY level shown here) as well as stochastic models (see Figs. 8, 9; itself are remarkably invariant to environmental variation mean population size not explicitly shown here) and also and observation error in the simulations performed. Here, in ciliate experiments (Fryxell et al. 2005). An escapement threshold harvesting outperforms both proportional and level set at the carrying capacity may also be attractive constant-quota harvesting with respect to average yield (and for controlling pest species that erupt due to unstable also yield constancy) in the presence of large environmental dynamics. In a comparison of six control strategies to reduce stochasticity and observation error. fluctuations, simulations by Tung et al. (2014)showed Such threshold values are well within the range that threshold harvesting with a threshold larger than the recommended in the fisheries literature (see Deroba and carrying capacity leads to the lowest population sizes when Bence 2008, Sect. 6.4). However, in this range of threshold compared to the other control strategies, while removing the values, the population variability basically vanishes, which largest harvest for highly chaotic populations. makes stock assessment more difficult (Hilborn and Walters If management aims at reducing a pest population size 1992). The strategy of proportional threshold harvesting as much as possible, or if the pest dynamics is stable, then adopts many features of threshold harvesting, but allows the thresholds far below the carrying capacity are advisable, but population size to vary over a wider range (Engen et al. they come at the cost of almost always having to harvest, 1997; Hilker and Liz 2019). in both deterministic and stochastic models. The fruit fly Moreover, there is the need and difficulty of knowing experiments by Tung et al. (2016) implemented such low the population size for applying threshold harvesting. threshold values, but they found almost no effect on the Measurement error and uncertainty have been shown to average population size. However, their population census change the optimal strategy for economic exploitation (e.g., was taken just before harvesting and is therefore affected by Clark and Kirkwood 1986; Sethi et al. 2005; Deroba and increased population growth after reduced due Bence 2008) and to exacerbate the biological problem of to harvesting (a “hidden” hydra effect, see Hilker and Liz (Roughgarden and Smith 1996). Haydon and 2013). Fryxell (2004) show that reduced uncertainty by increased If threshold harvesting is used as an exploitation strategy, understanding and prediction of population variation will the maximum sustainable yield is achieved at thresholds have only a modest effect on absolute yield, but may far below the carrying capacity. On the experimental side, substantially reduce the risk of overexploitation. neither the ciliate (Fryxell et al. 2005) nor the fruit fly All these results suggest that the choice of threshold microcosms (Tung et al. 2016) showed evidence for the values or management strategy in the first place depends on Theor Ecol (2020) 13:519–536 533 a suite of socio-economic, ecological, and political factors. Since g(x) = min{f(x),T}, it is clear that g(x) > x for Many comparisons of harvesting strategies are based on all x ∈ (0,T),andg(x) ≤ TT. Thus, T certain objectives such as maximizing yield, sometimes is the unique fixed point of g in (0, ∞). Choose an initial n also taking into account other factors (cf. Milner-Gulland condition x0 > 0. It is evident that g (x0) ≤ T

Fig. 10 Plots of the graph y = f 2(x) = f(f(x))of the ab second iteration of f (blue curve), and the line y = x (black). A: f(x)= xe1.885(1−x) is an SU-map. B: f(x)= 7.86x − 23.31x2 + 28.75x3 − 13.30x4 is unimodal but not an SU-map 534 Theor Ecol (2020) 13:519–536 equilibrium K = 1 with f (K) =−0.885, and hence, it is is composed of two minimal permutations of length 4, and asymptotically stable. Since f is an SU-map, K is the only 12345678 fixed point of f 2, and therefore, it is a global attractor of f π 4 = 8 21436587 (Fig. 10a). However, the unimodal map f :[0, 1]→[0, 1] defined by f(x) = 7.86x − 23.31x2 + 28.75x3 − 13.30x4 is composed of four minimal permutations of length 2. has a unique positive equilibrium K ≈ 0.726, with f (K) ≈ A cycle of f is of minimal type if its associated −0.885, as in the previous example. But f is not an SU- permutation is minimal. By Theorem 3.6 in Sharkovsky map and f 2 has five positive fixed points (Fig. 10b). Of et al. (1997), if a map has only cycles whose periods are a course, K is not a global attractor for f in this case, since power of 2, then all cycles are of minimal type. These maps an attracting 2-cycle coexists with the attracting fixed point. are called simple maps, and their cycles are referred to as This polynomial has been used in Singer (1978) to illustrate simple cycles. the importance of the Schwarzian derivative. The properties of simple cycles are responsible for A more general result establishes that SU-maps satisfy- the fine bifurcation structure of Eq. (2.2). For example, ing condition (A) have at most one attracting cycle (see if {x1,x2,...,x8} is the 8-cycle associated to π8,then 4 4 Theorem 5.1 and Corollary 5.9 in Sharkovsky et al. 1997). f (x7) = x8 and f (x8) = x7. Thus, there exists y4 ∈ 4 Assuming that the map f has only a finite number of peri- (x7,x8) such that f (y4) = y4. Moreover, y4 = max{x> odic orbits, Theorem 5.3 in the same reference ensures that 0 : f 4(x) ≥ x}. N the attracting cycle has period 2 , for an integer N ≥ 0, Since the 4-cycle {y1,y2,y3,y4} is simple, it follows that i 2 2 and f has only cycles of period 2 , with 0 ≤ i ≤ N.Fur- f (y3) = y4 and f (y4) = y3. Therefore, there exists 2 thermore, the properties of simple cycles (Sharkovsky et al. z2 ∈ (y3,y4) such that f (z2) = z2. Moreover, z2 = 1997, Chapters 3 and 4) guarantee the simple structure of max{x>0 : f 2(x) ≥ x}. Finally, there exists a fixed point the period-halving bifurcations that occur as T is decreased: K ∈ (z1,z2),wherez1 = f(z2),andK = max{x>0 : an attracting cycle {x1,x2,...,xp} (x1

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